Columns are related to Fibonacci n-step numbers. Are there closed forms for the sequences in the columns?

We denote by a(n,k) the number which is in the (n+1)-th row and (k+1)-th-column. With help of the definition, we have also the recurrence relation: a(n+k+1,k)=2*a(n+k,k)-a(n,k). We see on the main diagonal the numbers 1,2,4, 8, ..., which is clear from the formula for the general term d(n)=2^n. - Richard Choulet, Jan 31 2010

The general term in the n-th row and k-th column is given by: a(n,k) = Sum_{j=0..floor(n/(k+1))} ((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)). For example: a(5,3) = binomial(5,5)*2^5 - binomial(2,1)*2^1 = 28. The generating function of the (k+1)-th column satisfies: psi(k)(z)=1/(1-2*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1-z)). - Richard Choulet, Jan 31 2010

EXAMPLE

Triangle begins:

n\k|....0....1....2....3....4....5....6....7....8....9...10

---|-------------------------------------------------------

0..|....1

1..|....1....1

2..|....1....2....1

3..|....1....3....2....1

4..|....1....4....4....2....1

5..|....1....5....7....4....2....1

6..|....1....6...12....8....4....2....1

7..|....1....7...20...15....8....4....2....1

8..|....1....8...33...28...16....8....4....2....1

9..|....1....9...54...52...31...16....8....4....2....1

10.|....1...10...88...96...60...32...16....8....4....2....1

MAPLE

for k from 0 to 20 do for n from 0 to 20 do b(n):=sum((-1)^j*binomial(n-k*j, n-(k+1)*j)*2^(n-(k+1)*j), j=0..floor(n/(k+1))):od: seq(b(n), n=0..20):od; # Richard Choulet, Jan 31 2010